Completely agree with all your reasons about why they wouldn't time the essay questions/multiple choice.
Although, technically, if they were screening the application first then they could provide some sort of unique identifier so that you had to log in before you could see/answer the questions. Gets a bit messy though.
I had a look on the following page ;
British Airways Future Pilot Programme | FTEJerez
and I think this is the section that seems to say what the OP is referring to ;
Once your application has been assessed and accepted, you will progress to Phase II of the selection process. Phase II consists of a series of questions that you will be required to answer within a given time. If successful you will progress to Phase III which is an assessment day at a venue (soon to be confirmed) in London, UK, which includes the following tests: